Chapter 3

1. Chapter 3 Vectors and Two-Dimensional Motion

2. Vector vs. Scalar Review • All physical quantities encountered in this text will be either a scalar or a vector • A vector quantity has both magnitude (size) and direction • A scalar is completely specified by only a magnitude (size)

3. Vector Notation • When handwritten, use an arrow: • When printed, will be in bold print with an arrow: • When dealing with just the magnitude of a vector in print, an italic letter will be used: A

4. Properties of Vectors • Equality of Two Vectors • Two vectors are equal if they have the same magnitude and the same direction • Movement of vectors in a diagram • Any vector can be moved parallel to itself without being affected

5. More Properties of Vectors • Negative Vectors • Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) • Resultant Vector • The resultant vector is the sum of a given set of vectors

6. Adding Vectors • When adding vectors, their directions must be taken into account • Units must be the same • Algebraic Methods • More convenient

7. Notes about Vector Addition • Vectors obey the Commutative Law of Addition • The order in which the vectors are added doesn’t affect the result

8. Vector Subtraction • Special case of vector addition • Add the negative of the subtracted vector • Continue with standard vector addition procedure

9. Multiplying or Dividing a Vector by a Scalar • The result of the multiplication or division is a vector • The magnitude of the vector is multiplied or divided by the scalar • If the scalar is positive, the direction of the result is the same as of the original vector • If the scalar is negative, the direction of the result is opposite that of the original vector

10. Components of a Vector • A component is a part • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes

11. Example 1 Find the components of the following vector if it has a magnitude of 300m/s and the vector makes a 37o angle with the x-axis.

12. θ=400 500m Example 2 Find the components of the following vector.

13. Components of a Vector, cont. • The x-component of a vector is the projection along the x-axis • The y-component of a vector is the projection along the y-axis • Then,

14. More About Components of a Vector • The previous equations are valid only if θ is measured with respect to the x-axis • The components can be positive or negative and will have the same units as the original vector

15. More About Components, cont. • The components are the legs of the right triangle whose hypotenuse is

16. Adding Vectors Algebraically • Choose a coordinate system and sketch the vectors • Find the x- and y-components of all the vectors • Add all the x-components • This gives Rx:

17. Adding Vectors Algebraically, cont. • Add all the y-components • This gives Ry: • Use the Pythagorean Theorem to find the magnitude of the resultant: • Use the inverse tangent function to find the direction of R:

18. Signs of Components • 1st Quadrant • x-component is positive • y-component is positive

19. Signs of Components • 2nd Quadrant • x-component is negative • y-component is positive

20. Signs of Components • 3rd Quadrant • x-component is negative • y-component is negative

21. Signs of Components • 4th Quadrant • x-component is positive • y-component is negative

22. Example 3 Each of the displacement vectors A and B shown in the figure has a magnitude of 3.00 m. Find the magnitude and the direction of 4A + 5B by components method.

23. Motion in Two Dimensions • Using + or – signs is not always sufficient to fully describe motion in more than one dimension • Vectors can be used to more fully describe motion • Still interested in displacement, velocity, and acceleration

24. Displacement • The position of an object is described by its position vector, • The displacement of the object is defined as the change in its position

25. Example 4 A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal. Next, it travels 135 ft at an angle of 40.0° below the horizontal.

26. Example 5 Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x-axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x-axis. Find (a) the vector sum A + B and (b) the vector difference A – B.

27. Example 6 A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

28. Example 7 An airplane starting from airport A flies 300 km east, then 350 km at 30.0° west of north, and then 150 km north to arrive finally at airport B. (a) The next day, another plane flies directly from A to B in a straight line. In what direction should the pilot travel in this direct flight? (b) How far will the pilot travel in the flight? Assume there is no wind during either flight.

29. The average velocity is the ratio of the displacement to the time interval for the displacement The instantaneous velocity is the limit of the average velocity as Δt approaches zero The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion Velocity

30. Acceleration • The average acceleration is defined as the rate at which the velocity changes • The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero

31. Ways an Object Might Accelerate • The magnitude of the velocity (the speed) can change • The direction of the velocity can change • Even though the magnitude is constant • Both the magnitude and the direction can change

32. Projectile Motion • An object may move in both the x and y directions simultaneously • It moves in two dimensions • The form of two dimensional motion we will deal with is called projectile motion

33. Assumptions of Projectile Motion • We may ignore air friction • We may ignore the rotation of the earth • With these assumptions, an object in projectile motion will follow a parabolic path

34. Rules of Projectile Motion • The x- and y-directions of motion are completely independent of each other • The x-direction is uniform motion • ax = 0 • The y-direction is free fall • ay = -g • The initial velocity can be broken down into its x- and y-components

35. Projectile Motion

36. Projectile Motion at Various Initial Angles • Complementary values of the initial angle result in the same range • The heights will be different • The maximum range occurs at a projection angle of 45o

37. Some Details About the Rules • x-direction • ax = 0 • x = vxot • This is the only operative equation in the x-direction since there is uniform velocity in that direction

38. More Details About the Rules • y-direction • free fall problem • a = -g • take the positive direction as upward • uniformly accelerated motion, so the motion equations all hold

39. Velocity of the Projectile • The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point • Remember to be careful about the angle’s quadrant

40. Problem-Solving Strategy • Select a coordinate system and sketch the path of the projectile • Include initial and final positions, velocities, and accelerations • Resolve the initial velocity into x- and y-components • Treat the horizontal and vertical motions independently

41. Problem-Solving Strategy, cont • Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile • Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile

42. Some Variations of Projectile Motion • An object may be fired horizontally • The initial velocity is all in the x-direction • vo = vx and vy = 0 • All the general rules of projectile motion apply

43. Non-Symmetrical Projectile Motion • Follow the general rules for projectile motion • Break the y-direction into parts • up and down • symmetrical back to initial height and then the rest of the height

44. Example 8 An artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. To clear an avalanche, it explodes on a mountainside 42.0 s after firing. What are the x- and y-coordinates of the shell where it explodes, relative to its firing point?

45. Example 9 A child kicks a rock off the top of a 30m cliff, if the rock lands 4.2m from the base of the cliff, what horizontal velocity was given to the rock?

46. Example 10 Tom the cat is chasing Jerry the mouse across the surface of a table 1.5 m above the floor. Jerry steps out of the way at the last second, and Tom slides off the edge of the table at a speed of 5.0 m/s. Where will Tom strike the floor, and what velocity components will he have just before he hits?

47. Example 11 An rescue plane drops a package of emergency rations to stranded hikers. The plane is traveling horizontally at 60.0 m/s at a height of 300 m above the ground. Assume the hikers are 200 meters away from where the package was released (a horizontal distance) and the package landed somewhere in front of them. How far do the hikers have to walk to pick up the package from the ground?

48. Example 12 A home run is hit in such a way that the baseball just clears a wall 21 m high, located 130 m from home plate. The ball is hit at an angle of 35° to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume that the ball is hit at a height of 1.0 m above the ground.)

49. Relative velocity is about relating the measurements of two different observers It may be useful to use a moving frame of reference instead of a stationary one It is important to specify the frame of reference, since the motion may be different in different frames of reference There are no specific equations to learn to solve relative velocity problems Relative Velocity

50. Example 13 A jet airliner moving initially at 300 mi/h due east enters a region where the wind is blowing at 100 mi/h in a direction 30.0° north of east. What is the new velocity of the aircraft relative to the ground?